 |
(1)
|
其中
 |  | ![x+[gamma_1h_1(x)]+[gamma_2h_2(x)+gamma_1^2h_(11)(x)]+[gamma_3h_3(x)+gamma_1gamma_2h_(12)(x)+gamma_1^3h_(111)(x)]+[gamma_4h_4(x)+gamma_2^2h_(22)(x)+gamma_1gamma_3h_(13)(x)+gamma_1^2gamma_2h_(112)(x)+gamma_1^4h_(1111)(x)]+...,](/images/equations/Cornish-FisherAsymptoticExpansion/Inline3.svg) |
(2)
|
其中
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  | ![-1/(36)[2He_3(x)+He_1(x)]](/images/equations/Cornish-FisherAsymptoticExpansion/Inline12.svg) |
(5)
|
 |  |  |
(6)
|
 |  | ![-1/(24)[He_4(x)+He_2(x)]](/images/equations/Cornish-FisherAsymptoticExpansion/Inline18.svg) |
(7)
|
 |  | ![1/(324)[12He_4(x)+19He_2(x)]](/images/equations/Cornish-FisherAsymptoticExpansion/Inline21.svg) |
(8)
|
 |  |  |
(9)
|
 |  | ![-1/(384)[3He_5(x)+6He_3(x)+2He_1(x)]](/images/equations/Cornish-FisherAsymptoticExpansion/Inline27.svg) |
(10)
|
 |  | ![-1/(180)[2He_5+3He_3(x)]](/images/equations/Cornish-FisherAsymptoticExpansion/Inline30.svg) |
(11)
|
 |  | ![1/(288)[14He_5(x)+37He_3(x)+8He_1(x)]](/images/equations/Cornish-FisherAsymptoticExpansion/Inline33.svg) |
(12)
|
 |  | ![-1/(7776)[252He_5(x)+832He_3(x)+227He_1(x)]](/images/equations/Cornish-FisherAsymptoticExpansion/Inline36.svg) |
(13)
|
以及 埃尔米特多项式 的组合。
Cornish-Fisher 渐近展开
另请参阅
查理耶级数, 埃奇沃思级数使用 Wolfram|Alpha 探索
参考文献
Abramowitz, M. 和 Stegun, I. A. (编). 数学函数手册，包含公式、图表和数学表格，第 9 版。 纽约: Dover, p. 935, 1972.Cornish, E. A. 和 Fisher, R. A. "分布规范中的矩和累积量。" 国际统计学会评论 4, 1-14, 1937. 重印于 Fisher, R. A. 数理统计贡献。 纽约: Wiley, 1950.Wallace, D. L. "分布的渐近近似。" 数学统计年鉴 29, 635-654, 1958.Wasow, W. "关于某些分布到正态分布的渐近变换。" 应用数学研讨会论文集 VI，数值分析。 纽约: McGraw-Hill, pp. 251-259, 1956.在 Wolfram|Alpha 中被引用
Cornish-Fisher 渐近展开引用为
Weisstein, Eric W. "Cornish-Fisher 渐近展开。" 来自 MathWorld--Wolfram Web 资源。 https://mathworld.net.cn/Cornish-FisherAsymptoticExpansion.html